DOKTORS DER NATURWISSENSCHAFTEN
Karlsruhe
genehmigte
DISSERTATION
von
Referent: Prof. Dr. Friedemann Wenzel
Korreferent: Prof. Dr. Serge A. Shapiro
Universitat Karlsruhe (TH)
Germany
Abstract
Numerical modelling of seismic waves in heterogeneous, porous reservoir rocks is
an important tool for the interpretation of seismic surveys in reservoir engineer-
ing. Computer simulations allow the assessment of seismic scattering estimates
in heterogeneous environments as well as acoustic attenuation caused by wave-
induced flow of pore fluids. Furthermore, there are various theoretical studies
that derive effective elastic moduli and seismic attributes from complex rock
properties, involving patchy saturation and fractured media. In order to confirm
and further develop rock physics theories for reservoir rocks, accurate numerical
modelling tools are required.
In this thesis, a 2-D velocity-stress finite-differences (FD) scheme is presented
that allows to simulate waves within poroelastic media as described by Biot the-
ory. The scheme is second-order in time, contains higher-order spatial derivative
operators and is parallelised using the domain decomposition technique. Nu-
merical stability and dispersion relations of explicit poroelastic FD methods are
reviewed and these relations are exemplified by a series of numerical tests that
are compared to exact analytical solutions. The focus of several numerical ap-
plications is on accurate modelling of scattering and wave-induced flow in the
vicinity of mesoscopic heterogeneities such as cracks and gas inclusions. In or-
der to extract seismic attenuation and dispersion from quasistatic experiments,
the FD experiments are complemented by numerical experiments based on the
finite-element method.
The results confirm that finite-difference and finite-element modelling are
valuable tools to simulate wave propagation and coupled diffusion in heteroge-
neous poroelastic media, provided that the temporal and spatial scales not only
of the propagating waves but also of the induced fluid diffusion processes are
resolved properly.
Die numerische Modellierung von seismischen Wellen in porosen Reservoirgestei-
nen ist ein wichtiges Werkzeug fur die Interpretation von seismischen Daten und
von gesteinsphysikalischen Labormessungen. Mit Hilfe von Computerberechnun-
gen lassen sich fur komplexe, heterogene Gesteine die seismische Streudampfung
ebenso ermitteln wie Abschatzungen der akustischen Dampfung infolge von wel-
leninduzierten Fluidbewegungen. Zudem gibt es eine Vielzahl an theoretischen
Modellen, die effektive elastische Eigenschaften und seismische Attribute hete-
rogener Gesteine quantifizieren, beispielsweise im Falle von teilsaturierten oder
geklufteten Medien. Um theoretischen Modelle zu uberprufen, weiterzuentwickeln
und um die Grenzen ihrer Anwendbarkeit zu untersuchen, sind genaue Compu-
termodelle notwendig.
Ziel und Motivation dieser Arbeit ist es, einen Uberblick zu geben uber theo-
retische, gesteinsphysikalische Modelle, die die Wellenausbreitung in porosen Me-
dien beschreiben. Zudem wird ein neues 2-D Finite-Differenzen-Verfahren (FD)
entwickelt, das die Simulation der Wellenausbreitung in poroelastischen Medi-
en ermoglicht. Dem Verfahren liegt die Biot-Theorie zu Grunde, die neben der
Wellenausbreitung auch quasistatische Konsolidierungsprozesse beschreibt. Diese
stehen in engem Zusammenhang mit der Dispersion und Dampfung seismischer
Wellen infolge von mesoskopischen Prozessen an internen Heterogenitaten, etwa
Kluften oder Gaseinschlussen. Fur solche quasistatischen Prozesse werden die FD
Berechnungen durch numerische Experimente erganzt, die auf der Methode der
Finiten Elemente (FE) basieren.
Porenraumen bestehen. Ihre gesteinsphysikalischen Eigenschaften ergeben sich
folglich aus der Eigenschaften des Porenfluides und der Gesteinskorner, sowie de-
ren Anordnung auf Porenskala. Diese sogenannte Mikrostruktur wird ublicherwei-
se mit Hilfe von petrologischen Parameter wie der Porositat φ, der Permeabilitat
κ oder der Porenraumtortuositat ν charakterisiert. Neben mikroskaligen Hetero-
iii
iv
die etwa als Schichtgrenzen in seismischen Messungen sichtbar werden. Struktu-
ren, die von seismischen Wellen aufgelost werden, bezeichnet man als makroskalig.
Eine dritte Skala, die Mesoskala, umfasst hingegen all jene Strukturen, die zwar
kleiner sind als die seismische Wellenlange, jedoch deutlich großer als die Po-
renraumskala. Der Umstand, dass verschiedene geophysikalische Effekte auf allen
beschriebenen Skalen stattfinden, erklart die Komplexitat des Materialverhaltens
von Reservoirgesteinen.
von 10 bis 100Hz sind besonders die Wellenstreuung an Mediumsheterogenitaten
relevant, sowie mikro- und mesoskopische Porenfluidstromungen, welche durch
die einfallenden Wellen induziert werden konnen. Letztere fuhren durch viskose
Reibung zwischen Porenfluid und Korngerust zur Energiedissipation und damit
zu einer charakteristischen Wellendampfung, die in Feldmessungen beobachtbar
ist.
neben der viskosen Reibung auch Tragheitseffekte die Porenfluidstromungen, so
dass in porosen Medien neben den fur elastischen Materialien bekannten Kom-
pressions- und Scherwellen eine zweite, langsame Kompressionswelle auftreten
kann. Diese von Biot (1956a) theoretisch vorausgesagte langsame Wellenmode
wurde von Plona (1980) anhand von Ultraschallmessungen in einem kunstlichen
porosen Material hoher Porositat experimentell bestatigt. Aktuellere Forschung
im Bereich der experimentellen Gesteinsphysik widmet sich zunehmend der Quan-
tifizierung von mesoskopischen Effekten, wobei neben Ultraschallmessungen auch
bildgebende Verfahren der Computertomografie Anwendung finden. Dabei mo-
tivieren die immer detaillierteren Laborergebnisse neben der Weiterentwicklung
von theoretischen Erklarungsmodellen auch den starkeren Einsatz numerischer
Verfahren zur Simulation und Interpretation der im Labor gemessenen Daten.
Mathematische Modelle der Wellenausbreitung
Die erste vollstandige Theorie der dynamischen Poroelastizitat wurde von Mau-
rice Biot (1956a,b) entwickelt. Sie beschreibt die Wellenausbreitung elastischer
Wellen in porosen Medien in der Form von zwei gekoppelten Wellengleichungen
ρbu + ρfw = ∇[(λu + µ)∇.u + αM ∇.w] + µ∇2u , (1)
ρf u + Y ∗ w = ∇[αM ∇.u +M ∇.w] (2)
fur die zwei Verschiebungsfelder u und w. Hierbei bezeichnen ρb und ρf die
Gesamt- bzw. die Fluiddichte, λu, µ, α und M sind poroelastische Materialpa-
v
rameter. Reibungseffekte zwischen Fluid und Matrix werden durch den viskody-
namischen Operator Y erfasst. Fundamentallosungen der Biot-Gleichungen sind
drei ebene Wellenmoden, von denen zwei den aus der Elastomechanik bekann-
ten Kompressions- und Scherwellen entsprechen. In porosen Medien sind diese
Wellen leicht dispersiv mit einer charakteristischen Ubergangsfrequenz ωB, der
Biot-Frequenz. Diese liegt typischerweise im Bereich von 100kHz bis 1MHz, so
dass der Dispersionseffekt bei seismischen Messungen nicht in Erscheinung tritt.
Die dritte Wellenmode, langsame Kompressionswelle oder PII-Welle genannt,
ist im seismischen Frequenzbereich sehr stark gedampft und verhalt sich praktisch
rein diffusiv. Mithilfe der quasistatischen Approximation lasst sich die langsame
P -Welle daher naherungsweise als Diffusionswelle beschreiben. Erst bei hohen
Frequenzen im Bereich der Biot-Frequenz zeigt sie den Charakter einer propagie-
renden Welle.
Gegenstand intensiver gesteinsphysikalischer Forschung. Dabei besteht ein be-
sonderes Interesse an der mechanischen Beschreibung teilsaturierter und/oder
geklufteter Gesteine. Zu den klassischen Ansatzen zahlt z. B. das Modell von
White et al. (1975), mit dem der Einfluss von Gasinklusionen definierter Geome-
trie (Kugel, Schicht) auf die Dispersion und Dampfung von seismischen Wellen be-
schrieben wird. In neueren Modellen wird hingegen die Heterogenitat des Medium
nicht durch eine bestimmte Geometrie charakterisiert, sondern durch eine statisti-
sche Verteilungsfunktion der Materialparamter (Gurevich and Lopatnikov, 1995;
Muller and Gurevich, 2005a). Die effektive Wellenzahl des heterogenen Mediums
wird dabei durch die Methode der statistischen Glattung (engl. method of statis-
tical smoothing) der zufallsverteilten Materialparameter gewonnen.
Ein neuer Aspekt, der in dieser Arbeit behandelt wird, ist die Anwendung zu-
fallsbasierter Modelle auf den Fall der von Karman Verteilungsfunktion. Mit dem
Modell lassen sich teilsaturierte Medien beschreiben, deren Fluidphasen fraktal
verteilt sind.
Numerische Methoden
Falls ein poroelastisches Problem analytisch nicht losbar ist, so kann mit Hilfe nu-
merischer Verfahren eine Naherungslosung bestimmt werden. Im Kontext dieser
Arbeit werden zwei Verfahren verwendet: das Finite-Differenzen-Verfahren zur
Losung der dynamischen Biot-Gleichungen sowie die Methode der Finiten Ele-
mente fur rein quasistatische Fragestellungen. Der Schwerpunkt liegt dabei auf
dem FD Verfahren, da es im Rahmen dieser Arbeit entwickelt wurde, wahrend fur
die FE Berechnungen das kommerzielle Softwarepaket Abaqus verwendet wird.
vi
Zur Entwicklung des FD Verfahrens werden zunachst die gekoppelten Wel-
lengleichungen 1 und 2 in Form von vier Entwicklungsgleichungen erster Ord-
nung formuliert, wobei die verwendeten Feldgroßen die Partikelgeschwindigkeit
des porosen Mediums, die Filtrationsgeschwindigkeit des Porenfluides, der Ge-
samtspannungstensor sowie der Porendruck sind. Die zeitliche Diskretisierung er-
folgt durch das Ersetzen der zeitlichen Ableitungsoperatoren durch zentrale finite
Differenzen. Durch zeitliche Staffelung der Diskretisierung von Geschwindigkei-
ten und Spannungen wird ein Verfahrensfehler zweiter Ordnung erreicht. Analog
zur zeitlichen Diskretisierung erfolgt auch die raumliche Diskretisierung mit Hil-
fe zentraler FD-Operatoren. Dabei kommen raumliche Operatoren hoherer Ord-
nung zum Einsatz, wahlweise in klassischer Form oder auf einem gedrehten Git-
ter (Saenger et al., 2000). Da ein expliziter Zeitschrittoperator verwendet wird,
liefert das poroelastischen FD-Verfahren nur unter der Bedingung eines ausrei-
chend kleinen Zeitschrittes stabile Ergebnisse, wobei die Stabilitatseigenschaften
vergleichbar sind mit denen konventioneller Verfahren fur elastische Wellenaus-
breitung. Allerdings muss, um stabile Ergebnisse zu erhalten, als zusatzliche Be-
dingung ν/φ > ρf/ρb gewahrleistet sein. Fur die Berechnung großer Modelle ist es
schließlich vorteilhaft, das FD-Verfahren parallel auf einem Großrechner durchzu-
fuhren, was durch die Technik der Gebietszerlegung (engl. domain decomposition)
erreicht wird.
In einem zweiten Abschnitt dieses Kapitels uber numerische Methoden wird
eine Ubersicht uber die Methode der Finiten Elemente fur poroelastische Frage-
stellungen gegeben. Im Unterschied zum FD-Verfahren werden hierfur nicht die
Differenzialoperatoren diskretisiert sondern der zu Grunde liegende Losungsraum.
Dieser wird bei der FE Methode durch Polynom-Ansatzfunktionen mit ortlich be-
grenztem Trager dargestellt. Durch Multiplikation mit den Ansatzfunktionen und
Integration uber den Losungsraum gehen die Grundgleichungen in ein lineares al-
gebraisches System uber, das mit Hilfe eines Gradientenabstiegsverfahren gelost
wird. Zu beachten ist, dass die Implementation des Abaqus FE Programms le-
diglich die quasistatischen Biot-Gleichungen lost und somit dynamische Effekte
wie Wellenausbreitung nicht modelliert werden konnen.
Genauigkeits- und Skalierungstests
Ein entscheidender Aspekt bei der Berechnung der Wellenausbreitung mit dem
FD-Verfahren ist die Genauigkeit der numerischen Naherungslosung. Durch den
Vergleich von numerischen Ergebnissen mit exakten, analytischen Losungen kann
die Genauigkeit des FD-Verfahrens untersucht werden, was Gegenstand dieses
Kapitels ist.
Ebenso wie bei Modellierung elastischer Wellen ist es im poroelastischen Fall
vii
notwendig, die Wellenlangen zeitlich und raumlich ausreichend genau aufzulosen,
um den numerische Dispersionsfehler zu begrenzen. Dabei ist die langsamste Wel-
lenmode entscheidend, d. h. es muss die Diffusionswellenlange aufgelost werden,
um genaue Ergebnisse zu erzielen. Der Umstand, dass bei Frequenzen weit un-
terhalb der Biot-Frequenz die Skala des Diffusionsprozesses weitaus kleiner ist als
die die Wellenlange der schnellen Kompressionswelle, wird als numerische Stei-
figkeit (engl. numerical stiffness) bezeichnet. Diese fuhrt insbesondere bei der
Berechnung von poroelastischen Wellen im seismischen Frequenzbereich zu er-
heblichem Rechenaufwand, was anhand des Beispiels einer Reflektion von einer
poroelastischen Grenzflache gezeigt wird.
Ferner behandelt dieses Kapitel auch die konsistente Modellierung von freien
Fluiden im Rahmen der Poroelastizitatstheorie sowie einen Test zur Bestimmung
der Skalierbarkeit des parallelen Codes. Dabei stellt sich heraus, dass die Effizienz
des Programms mit steigender Anzahl von bearbeitenden Prozessen abnimmt,
jedoch bei 64 parallelen Prozessen noch 91% der Effizienz eines seriellen Prozesses
erreicht wird.
Anwendungen
Ziel dieses Kapitels ist es, anhand numerischer Beispiele zu zeigen, wie poro-
elastische Modellierung einen Beitrag zur Losung aktueller gesteinsphysikalischer
Forschung leisten kann. Beginnend mit FD-Experimenten der Wellenstreuung an
einfachen poroelastischen Inklusionen lasst sich die unterschiedliche Wellenkon-
version an internen Grenzflachen veranschaulichen. Es werden Ergebnisse gezeigt
fur ein teilsaturiertes Medium sowie fur einen elliptischen Riss. Wenn die Wellen-
lange sehr groß ist im Verhaltnis zur untersuchten Inklusion, findet hauptsachlich
Konversion von schnellen Wellen zur langsamen Diffusionswelle statt, so dass
das Gesamtverhalten mit der quasistatischen Approximation beschrieben werden
kann. Dies ermoglicht die Verwendung der FE-Methode zur Durchfuhrung von
quasistatischen Relaxationsexperimenten, mittels derer effektive Materialeigen-
schaften eines heterogenen, poroelastischen Mediums bestimmt werden konnen.
Falls die Geometrie des untersuchten Modells effektiv vertikal transversale Isotro-
pie (VTI) aufweist, genugen drei Experimente, um den vollstandigen Tensor der
Relaxationsrate zu bestimmen. Durch Fouriertransformation erhalt man ferner
den komplexen, frequenzabhangigen Elastizitatstensor, aus dem sich Dispersion
und Dampfung aller VTI-Wellenmoden berechnet lassen. Numerische Losungen
werden fur ein geschichtetes Medium gewonnen sowie fur ein 3-D Medium mit
einer elliptischen Inklusion.
viii
Eine Moglichkeit, diese numerisch zu quantifizieren, bieten elastische FD Pro-
pagationsexperimente. Zu diesem Zweck wird die relative Amplitudenanderung
einer ebenen Kompressionswelle entlang ihres Laufweges durch ein zufallsver-
teiltes Medium statistisch ausgewertet. Die elastische Streudampfung ist dabei
proportional zur Varianz dieser Amplitudenanderung. Konkret wird anhand ei-
ner Serie von numerischen Experimenten ein anisotrop korreliertes Medium in
Abhangigkeit des Welleneinfallswinkels untersucht. Die Ergebnisse werden an-
schließend interpretiert auf der Grundlage von analytischen Abschatzungen der
elastischen Streudampfung.
Im Unterschied zur rein elastischen Streuung gibt es in zufallsverteilten poro-
elastischen Medien die Moglichkeit, dass in Abhangigkeit vom Frequenzbereich,
quasistatische Dampfung infolge welleninduzierter Fluidstromungen stattfindet
in Kombination mit poroelastische Streuung. Dieser Ubergang wird anhand ei-
nes teilsaturierten Mediums untersucht, wobei die Fluidphasen zufallsverteilt sind
und einer fraktalen Verteilungsfunktion unterliegen. Abschließend zeigt ein Bei-
spiel die erfolgreiche Anwendung des FD-Verfahrens zur Simulation einer im La-
bor durchgefuhrten Ultraschallmessung an einem teilsaturierten Sandstein.
Schlussfolgerungen und Ausblick
Hauptursachen fur seismische Dampfung in geologischen Reservoiren. In dieser
Arbeit wird ein Uberblick gegeben uber die mathematischen Modelle zur Be-
schreibung der genannten Effekte auf die Wellenausbreitung in porosen Gestei-
nen, wobei insbesondere auch ein neuer Ansatz vorgestellt wird, mit Hilfe dessen
welleninduzierte Stromungen in zufallsverteilten Fraktalen quantifiziert werden.
Das Hauptergebnis der vorliegenden Arbeit umfasst die Entwicklung, Imple-
mentierung und Validierung eines neuen Finite-Differenzen-Verfahrens zur Lo-
sung der dynamischen Biot-Gleichungen. Das Verfahren erlaubt die Simulation
der Wellenausbreitung in heterogenen, poroelastischen Strukturen in einem brei-
ten Frequenzbereich. Da die dynamischen Biot-Gleichungen bei niedrigen Fre-
quenzen eine hohe numerische Steifigkeit aufweisen, ergibt sich fur die Simulation
seismischer Wellen ein hoher Diskretisierungsaufwand, um gleichzeitig makrosko-
pisch propagierende Wellen und kleinskalige Diffusionsprozesse aufzulosen. Zur
Untersuchung mesoskopischer Prozesse ist es daher vorteilhaft, einen quasistati-
schen Finite-Elemente-Loser zu verwenden.
dellierung einen wertvollen Beitrag leisten kann zur Untersuchung von Wellen-
ausbreitung und gekoppelten Diffusionsprozessen in heterogenen, poroelastischen
Medien.
Danksagungen
Ich danke Prof. Friedemann Wenzel fur seine Bereitschaft, diese Arbeit als Haupt-
referent zu begleiten und Prof. Serge Shapiro fur die Ubernahme des Korreferats.
Fur die ausgezeichnete Betreuung gilt zudem mein besonderer Dank Tobias Mul-
ler, der mir wahrend der gesamten drei Jahre am GPI stets mit motivierendem
Interesse und vielen Hinweisen hilfreich zur Seite gestanden hat.
Fur eine gute Zusammenarbeit und Gedankenaustausch wahrend zahlreicher
Kaffeepausen bin ich all meinen Kollegen sehr verbunden, in besonderer Weise
Johannes, Markus, Nico, Tian, Tatiana sowie Gerardo und Miro.
Schließlich mochte ich auch Sophie danken, die mit liebevoller Unterstutzung
und ermutigenden Worten sehr zum Gelingen dieser Arbeit beigetragen hat.
ix
Contents
1.2 Seismic attenuation in fluid-saturated rocks . . . . . . . . . . . . 4
1.3 Experimental laboratory results . . . . . . . . . . . . . . . . . . . 9
1.4 Motivation and overview of this thesis . . . . . . . . . . . . . . . 12
2 Mathematical models for wave propagation in porous media 15
2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.5 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.7 Heterogeneous porous media . . . . . . . . . . . . . . . . . . . . . 29
2.8 Wave-induced fluid flow . . . . . . . . . . . . . . . . . . . . . . . 32
2.8.1 White’s model for partial saturation . . . . . . . . . . . . 33
2.8.2 Continuous random media models . . . . . . . . . . . . . . 35
2.9 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.1 Modelling poroelastic wave propagation using the FD method . . 42
3.1.1 Time discretisation . . . . . . . . . . . . . . . . . . . . . . 43
3.1.3 Boundaries and sources . . . . . . . . . . . . . . . . . . . . 48
3.1.4 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.1.5 Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.1.6 Parallelisation . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.2.1 Spatial discretisation by the Galerkin method . . . . . . . 54
x
3.2.3 Time integration and solution of the linear system . . . . . 56
3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.1 Numerical dispersion . . . . . . . . . . . . . . . . . . . . . . . . . 59
4.3 Resolution of the diffusion boundary layer . . . . . . . . . . . . . 62
4.4 Modelling free fluids . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.5 Parallel performance . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.2 Quasistatic relaxation experiments . . . . . . . . . . . . . . . . . 73
5.3 Elastic scattering in random media . . . . . . . . . . . . . . . . . 81
5.4 Wave-induced flow in random media . . . . . . . . . . . . . . . . 88
5.5 Simulation of ultrasonic laboratory experiments . . . . . . . . . . 96
5.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
A Viscoelasticity and quality factor 105
B Statistical characterisation of random media 109
C Supplementary rock physics formulas 113
C.1 High frequency correction for the Biot equations . . . . . . . . . . 113
C.2 Poroelastic Backus average . . . . . . . . . . . . . . . . . . . . . . 114
C.3 Extended theory of wave-induced flow in layered porous media . . 117
C.4 White’s model for partial saturation . . . . . . . . . . . . . . . . . 118
C.5 Complement on the random fractal media model . . . . . . . . . . 120
D Abaqus porous elastic model 123
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
1.1 Pore spaces of four different natural carbonate rocks. . . . . . . . 2
1.2 Scales of reservoir characterisation. . . . . . . . . . . . . . . . . . 3
1.3 Classification of different attenuation mechanisms. . . . . . . . . . 4
1.4 Intrinsic P -wave attenuation 1/Q of rocks at one test site. . . . . 5
1.5 Characteristic frequencies of five relaxation mechanisms . . . . . . 7
1.6 Classification of scattering phenomena. . . . . . . . . . . . . . . . 8
1.7 Sketch of Plona’s experimental setup. . . . . . . . . . . . . . . . . 10
1.8 Seismograms proving the existence the slow P -wave mode. . . . . 10
1.9 Attenuation measurements in partially saturated sandstone. . . . 11
1.10 Computer tomography scans of limestone during gas injection. . . 12
1.11 Ultrasonic velocities in a partially saturated rock sample. . . . . . 13
2.1 Overview of theoretical descriptions of porous media . . . . . . . 16
2.2 Sketches of three deformation experiments . . . . . . . . . . . . . 21
2.3 Biot theory: dispersion and attenuation . . . . . . . . . . . . . . . 24
2.4 Biot theory: quasistatic pore pressure response . . . . . . . . . . . 28
2.5 Biot theory: quasistatic dispersion and attenuation . . . . . . . . 29
2.6 Gassmann-Hill and Gassmann-Wood bounds . . . . . . . . . . . . 31
2.7 Propagation velocities in an effectively anisotropic porous medium 32
2.8 Comparison of three theories of partial saturation . . . . . . . . . 35
2.9 Realisations of random media and correlation functions. . . . . . . 37
2.10 Continuous random media model for partial saturation . . . . . . 38
3.1 Finite-difference operators and their amplitude spectra . . . . . . 47
3.2 Stencils of standard and rotated FD operators. . . . . . . . . . . . 47
3.3 Examples of boundary conditions for poroelastic wave simulation. 49
3.4 Domain of stability, limited by two conditions. . . . . . . . . . . . 51
3.5 System eigenvalues of fast and slow P -waves. . . . . . . . . . . . . 52
3.6 Sketch of the parallelisation by domain decomposition. . . . . . . 53
3.7 Example of a finite-element mesh. . . . . . . . . . . . . . . . . . . 54
3.8 Linear basis functions for a triangular element. . . . . . . . . . . . 55
4.1 Numerical dispersion in a 2-D frictionless, porous medium. . . . . 60
xii
4.2 Synthetic velocity and pore pressure seismograms. . . . . . . . . . 61
4.3 Pore pressure profiles for a “step” loading at x = 0. . . . . . . . . 62
4.4 Frequency-dependent reflection from a gas-water contact. . . . . . 63
4.5 Frequency-dependent reflection from a fluid-porous interface. . . . 65
4.6 Seismograms of wave scattering from a fluid-porous interface . . . 66
4.7 Scalability test of the poroelastic FD scheme. . . . . . . . . . . . 68
5.1 Scattering of a plane compressional wave from an elliptic crack. . 71
5.2 Same as figure 5.1, but for a circular gas inclusion. . . . . . . . . . 72
5.3 Dispersion, attenuation in patchy-saturated rock (1-D model). . . 75
5.4 Three deformation states used to obtain all 5 effective moduli. . . 76
5.5 Two model geometries for double porosity relaxation experiments. 77
5.6 Attenuation derived from double porosity relaxation experiments. 78
5.7 Relaxation functions obtained from double porosity experiments. . 79
5.8 Maximum wave attenuation as a function of incidence angle. . . . 80
5.9 Backscattering and random diffraction in 1-D and 2-D media. . . 82
5.10 Velocity models used for numerical scattering experiments. . . . . 84
5.11 Synthetic seismograms of scattered plane compressional waves. . . 84
5.12 Amplitude spectra of scattered wavefields in isotropic media. . . . 85
5.13 Log-amplitude variance of different angles of incidence φ. . . . . . 86
5.14 Log-amplitude variance for large propagation distances. . . . . . . 87
5.15 Synthetic saturation maps with fractal pore fluid distributions. . . 89
5.16 Pore pressure snapshots of wave propagation in the BRM model. . 90
5.17 Same as 5.16 but for the CRM model. (Clipping also as in 5.16) . 90
5.18 Quasistatic pore pressure relaxation in the BRM model. . . . . . . 92
5.19 Same as 5.18 but for the CRM model. (Clipping also as in 5.18) . 92
5.20 Dispersion, attenuation in fractal media derived from FD exp. . . 93
5.21 Same as 5.20 but for the CRM model. . . . . . . . . . . . . . . . 93
5.22 CT images of the Casino Otway Basin sandstone. . . . . . . . . . 96
5.23 Velocity-saturation relation for a rock with partial saturation. . . 97
5.24 Unrelaxed pore pressure distributions in the 3-D FE model. . . . 100
A.1 Relaxation and creep test for the evaluation of M(ω). . . . . . . . 107
B.1 Procedure of creating random media realisations. . . . . . . . . . 111
C.1 Backus averaging and corresponding maximum attenuation. . . . 118
C.2 Values of the Gaussian hypergeometric function 2F1(a, b; c;x) . . . 121
C.3 Fractal CRM realisations for different Hurst exponents. . . . . . . 122
C.4 CRM model for partial saturation with fractal geometry. . . . . . 122
D.1 Stress-strain relation of the Abaqus FE model. . . . . . . . . . . . 124
D.2 Triangular, quadrangular and tetrahedral linear elements. . . . . . 125
List of Tables
2.2 Material properties of fluids typically found in reservoirs . . . . . 26
2.3 Overview of rock physics theories for wave-induced fluid flow. . . . 34
3.1 Finite-differences coefficients . . . . . . . . . . . . . . . . . . . . . 46
4.1 Elapsed wall clock time for two sets of test simulations. . . . . . . 67
5.1 Material properties used in the numerical applications. . . . . . . 70
5.2 Petrophysical properties of the dry Casino Otway sandstone . . . 97
xiv
Nomenclature
Roman
b friction coefficient Pa s / m2
c also v, wave propagation velocity m/s
cijkl (short notation cIJ) elasticity tensor Pa
f frequency Hz
qi filtration velocity m/s
wi relative displacement (Eq. 2.13) m
B correlation function in random media
C = αM , poroelastic modulus Pa
D wave parameter in seismic scattering (Eq. 1.5)
D hydraulic diffusivity (Eq. 2.75) m2/s
G shear modulus Pa
K bulk modulus Pa
N inverse uniaxial storage Pa
P P -wave modulus Pa
Q quality factor
α Biot coefficient of effective stress (Eq. 2.39)
δ Dirac distribution (Eq. 2.10)
δij Kronecker symbol (Eq. 2.9)
ijk Levi-Civita symbol (Eq. 2.8)
ε volumetric strain (Eq. 2.15)
εij strain tensor (Eq. 2.14)
ζ local increment of fluid content (Eq. 2.16)
η fluid viscosity Pa s
κ hydraulic permeability m2
λ Lame parameter Pa
λ wave length m
ν tortuosity
ρ mass density kg/m3
τij stress tensor Pa
χ logarithmic amplitude relation (Eq. 5.26)
ψ relaxation function (see appendix A)
ψijkl relaxation tensor
ωc characteristic frequency of mesoscopic flow (Eq. 2.87) 1/s
Indices / superscripts
xvii
Operators
φi,j gradient of φi 1/m
∂tφ time differential operator 1/s
∂iφ spatial differential operator 1/m
Diφ discrete spatial differential operator 1/m
F{φ} Fourier transform
Abbreviations
Introduction
Seismic surveys and acoustic borehole measurements are routinely used in the
hydrocarbon exploration industry in order to obtain information about subsurface
geology. The first aim of seismic processing techniques is to construct images
of velocity and reflectivity distributions from recorded elastic wave fields. In a
second step, from the seismic data mechanical and petrophysical properties are
derived such as rock compressibility, porosity and information about the presence
or absence of fluids in the pore space. It is the domain of seismic rock physics
to establish physical relationships between these rock properties and the seismic
response (Dewar and Pickford, 2001).
In particular, the influence of pore fluids on seismic velocity and attenuation
has attracted increasing attention in the past, since wave-induced fluid flow is
considered to contribute mainly to measured signatures in porous reservoirs, e. g.
an interesting question in current rock physics research is concerned with the
estimation of permeability from seismic data. Based on the pioneering work of
Maurice A. Biot on wave propagation in porous media in the 1950’s, a large num-
ber of publications has appeared in the literature dealing with the theoretical and
experimental description of porous media acoustics. With the general availabil-
ity of computers, in the 1990s first attempts were made to solve numerically the
equations governing wave propagation and coupled flow processes and this field
of numerical rock physics is becoming more and more important for the interpre-
tation of experimental observations and for testing the validity and applicability
of new theoretical models.
It is the purpose of this thesis to present a new numerical scheme for solving
the dynamically coupled wave equations in porous media and to demonstrate
how numerical tools are successfully applied to current problems in rock physics
research.
1
2 Introduction
Figure 1.1: Pore spaces of four different natural carbonate rocks. Pho-
tographs of thin sections show an oolitic limestone (1), a sample with large
pores due to the dissolution of microfossils (2), a nummulite limestone (3)
and a totally dolomised oolitic limestone (4). From Bourbie et al. (1987).
1.1 Scales in porous media
Hydrocarbon reservoir rocks such as sandstones, shales and carbonates are porous
media with fluids filling the pore space between mineral grains. Physical proper-
ties of reservoir rocks are therefore determined by the properties of its constituents
and in as much by the distribution of porespace and grain matrix, referred to as
the rock microstructure. The sub-millimetre scale microstructure of four lime-
stones is depicted in Figure 1.1, showing the large variability of natural porespace
geometries. If for one particular rock, one had all information about the the distri-
bution of the pore space and about the properties of grains and fluid, in principle
one could infer the overall mechanical and hydraulic behaviour of the composi-
tion (Gueguen and Palciauskas, 1994). Obviously, this information is usually not
available in practice and it is convenient to describe microstructure and associate
1.1 Scales in porous media 3
Figure 1.2: Scales of reservoir characterisation ranging from microscopic
grain sizes (a) via several centimetres for mesoscale heterogeneities (b) up
to seismic wavelengths that are tens of metres (c).
microscale effects using measurable quantities, among the most important are
porosity φ (the volume fraction of the pore space), hydraulic permeability κ (the
ability the conduct fluids) and overall elastic moduli of the rock matrix. Another
parameter is the pore space tortuosity ν, describing the ratio of average flow
path length inside the pore channels of a given rock sample and the total sample
dimension.
Besides the complexity of the porespace, rocks are typically heterogeneous
on various scales, as shown in Figure 1.2. The scale that is resolved by seismic
waves is that of geological layers and reservoir structures. Typically this so-called
macroscale ranges from several centimetres at 10kHz sonic logging frequency up
to tens of metres at 100Hz surface seismic records.
Finally, a third intermediate spatial scale can be defined that is due to het-
erogeneity of the porous medium properties. These so-called mesoscale hetero-
geneities are smaller than the seismic wavelength but still much larger than the
dimensions of the microscopic pore space. Actually rocks always contain to some
extent heterogeneity that is not due to the grains and porespace but to other
features such as fractures, soft inclusion, embedded thin layers or different fluids
distributed on various scales.
It is the multiscale nature of Earth materials that explains their complexity
and the high variability of their physical properties. Seismic measurements that
are carried out using a particular frequency always contain information about a
specific scale. If for example results from sonic logging are interpreted on a larger
scale, one has to take into account scaling effects that are simply not included
in the measurement. This is done by upscaling techniques. From the modelling
point of view, the reasonable and successful application of theoretical models
and numerical rock physics tools requires a good understanding of the physical
processes on the various scales.
4 Introduction
1.2 Seismic attenuation in fluid-saturated rocks
If an initially dry rock sample is fully saturated with water, its compressibility is
reduced while shear stiffness is practically not affected. This static effect has been
quantified by Fritz Gassmann’s work “On elasticity of porous media” (Gassmann,
1951) and is widely applied for fluid substitution calculations. Another fundamen-
tal effect is time-dependent consolidation of geomaterials under a given loading.
Terzaghi and Frohlich (1936) found that the consolidation of clay is governed by
a diffusive pore pressure relaxation process. His results were later generalised by
Biot (1941) for the three-dimensional case. Biot further developed his theory in
order to include wave propagation effects Biot (1956a,b) and brought up the idea
that pore pressure relaxation may lead to dispersion and attenuation of seismic
waves.
Attenuation denotes all processes leading to a loss in seismic wave amplitude
except for geometrical spreading effects. In general, two classes of wave attenua-
tion can be distinguished (see Figure 1.3). On the one hand, intrinsic attenuation
is caused by non-elastic energy losses, meaning that a part of the wave energy
is transferred to heat by internal friction. Apparent attenuation, on the other
hand, occurs when the wave amplitude is reduced by the redistribution of wave-
field energy (e. g. due to elastic scattering). It is well-known that the attenuation
is a frequency-dependent effect and that it is linked to velocity dispersion by the
causality principle. An introduction to viscoelastic material behaviour and to the
quantitative description of attenuation is given in appendix A. Figure 1.4 gives an
example of a broad frequency-range measurement of seismic intrinsic attenuation
at the Imperial College test site, combining ultrasonic core measurements with
1.2 Seismic attenuation in fluid-saturated rocks 5
Figure 1.4: Intrinsic P -wave attenuation 1/Q as determined by
Sams et al. (1997) on rocks at the Imperial College test site at various
depths. VSP and sonic log estimates have been corrected for scattering
attenuation. After Pride et al. (2003).
sonic logs, crosswell and VSP data (Sams et al., 1997). The measured values of
the inverse quality factor 1/Q attain 0.1 and higher for the sonic logs and more
than 0.02 for all the measurements, thus indicating that on all scales, a signif-
icant amount of energy loss is observed. In the following, mechanisms causing
attenuation of seismic waves in reservoirs are discussed in more detail.
In a homogeneous porous medium, pore pressure differences may appear be-
tween the peaks and the troughs of a propagating compressional wave. The relax-
ation associated with pressure equilibration between these extrema is the global
flow mechanism described by Biot (1956a). Since the scale of the Biot global flow
is that of the wavelength, it is a macroscopic effect. The characteristic relaxation
frequency of the process is given by
ωB = η φ
κ νρf , (1.1)
where η is the dynamic fluid viscosity, φ is porosity, κ the hydraulic permeability,
ν the pore space tortuosity and ρf the fluid density. Its values are typically of
the order of 100 kHz up to the MHz range and therefore, at seismic frequencies
(usually much below 1kHz) pore pressure is always unrelaxed with respect to
global flow effects and Biot attenuation is negligible. By the way, an analogy
exists between the theory of poroelasticity and thermoelasticity, where unrelaxed
processes are called adiabatic (Norris, 1992). The Biot frequency ωB will be
discussed later in more detail since it separates two regimes that are characterised
6 Introduction
by friction-dominated diffusive fluid flow on the one hand and inertially driven
fluid flow on the other hand, see chapter 2.
A second, very efficient attenuation mechanism occurs if a porous medium
is heterogeneous on mesoscopic scales, i. e. on scales larger than the pore scale
but still smaller than the seismic wavelength. In this case, pore pressure differ-
ences appear not only on macroscopic scales, but also locally across each internal
interface. Therefore, the relaxation may occur due to local flow effects and its
characteristic frequency depends explicitly on the scale of the heterogeneity as
ωc = κN
η L2 . (1.2)
Here, N is a poroelastic modulus, introduced later in section 2.6 and L is a
characteristic spatial scale of the medium heterogeneity. The local flow mecha-
nism is often referred to as wave-induced fluid flow (e. g. Muller and Gurevich,
2005b). It is important to note that while the characteristic frequency ωB de-
creases with increasing permeability, the characteristic local flow frequency ωc
shows the opposite behaviour. A second remarkable point is that because of the
presence of multiscale heterogeneities in porous rocks, seismic attenuation due to
wave-induced flow affect a large frequency range and play a major role at seismic
frequencies. In this context, note the spatial scale dependence of equation 1.2.
Finally, from a modelling point of view it is important to mention that local flow
effects are completely described by the Biot theory.
An attenuation effect that is not included in Biot’s description of seismic wave
propagation, that is, however, considered to be very efficient is the squirt flow
first described by Mavko and Jizba (1991). The squirt flow is very similar to the
local flow described above, but it emphasises grain-scale heterogeneities and can
therefore be classified as a microscopic effect. Actually, reservoir rocks practically
always have microcracks, loose grain contacts or defects that are often subsumed
as soft porosity. During wave propagation, the soft pore space is squeezed and
since the fluid in the pores is viscous, this leads to energy dissipation and wave
attenuation. The frequency-dependence of the squirt flow has been quantified by
Dvorkin et al. (1995). In their model, the characteristic frequency depends on
microscopic crack scale R and its aperture h such that according to Pride et al.
(2003) one can write
η , (1.3)
where Kf denote the fluid bulk modulus and β = (h/R)2. Interestingly, ωsquirt
depends on the fluid viscosity η but not on the permeability κ, unlike in the
case of local flow. The reason for this is that squirt relaxation process occurs on
lengthscales not exceeding the grain size. The characteristic squirt-flow length
1.2 Seismic attenuation in fluid-saturated rocks 7
Figure 1.5: Comparison of characteristic relaxation frequencies as pre-
dicted by rock physics theories for typical rock and fluid parameters. Ar-
rows show the direction of change as the labeled parameter increases.
Adapted from Mavko et al. (1998).
is an additional parameter that is not related to material properties appearing
in the theoretical Biot model. In subsequent chapters, the squirt flow effect will
not be considered, but the focus will be on global and local flow effects that are
directly described by the Biot theory.
Figure 1.5 gives an overview of the frequency ranges on which the previously
discussed relaxation processes may occur. As can be seen from this figure, the
Biot global flow occurs typically at ultrasonic frequencies, while squirt flow and
wave-induced local flow may very well affect the seismic frequency range. Typ-
ical frequencies where wave scattering occurs are shown in Figure 1.5, as well.
Scattering and the corresponding apparent attenuation is briefly discussed.
In contrast to the aforementioned intrinsic attenuation mechanisms, the scat-
tering of seismic waves in an elastic medium is not based on absorption but on
the redistribution of wavefield energy. It is therefore called apparent attenuation.
The scattering of seismic waves is most efficient when the seismic wavelength λ
approximately equals the characteristic size of the elastic scatterer, a, and the
effects of scattering become increasingly important with increasing propagation
distance L. According to Aki and Richards (1980), scattering phenomena can be
classified using the dimensionless quantities ka and kL, where k = 2π/λ is the
wavenumber. An overview of the different scattering regimes is given in Figure
1.6. If ka is very small, the wavelength is much larger than the scale of the hetero-
8 Introduction
Figure 1.6: Scattering regimes classified by the products of wavenumber
k, and characteristic scale a or propagation path L, respectively. From
Mavko et al. (1998).
geneities and the medium behaves like an effective homogeneous medium where
scattering is negligible. The effective medium theory requires that the fractional
energy loss E/E is small, as well. On the other hand, if ka is large, the wave
propagates through a piecewise homogeneous medium. A critical frequency for
scattering processes is given by ka = 1 or alternatively
ωs = c
a , (1.4)
where H is the elastic modulus and ρ the density. The wave parameter D defined
as
is another dimensionless number parameter characterising the scattering regime.
It is used as indicator whether diffraction has a significant impact on the scat-
tered wavefield. For D < 1 wave diffraction is small and ray theory can be applied
for the wavefield description. The diffraction regime D > 1 is then furthermore
subdivided into the weak and strong scattering regimes, depending on whether
forward scattering is dominant (weak) or multiple scattering occurs (strong). De-
pending on the scattering regime, different theoretical wavefield approximations
are available (Wu and Aki, 1988; Sato and Fehler, 1998; O’Doherty and Anstey,
1.3 Experimental laboratory results 9
1971; Shapiro and Hubral, 1999; Muller and Shapiro, 2001). A more detailed de-
scription of weak wave scattering in random elastic media is given in section 5.3
together with a corresponding numerical experiment.
1.3 Experimental laboratory results
One of the main results of Biot theory is the existence of a second compressional
wave mode – the slow P -wave – in porous media. To put it simply, this slow
wave mode is associated with an out-of-phase movement of the fluid and the
solid phases, while fluid and solid move in phase during fast P -wave propagation.
This theoretically predicted wave mode has been first experimentally observed by
Plona (1980), who carried out ultrasonic laboratory measurements on a synthetic
highly-porous medium consisting of sintered glass beads.
Synthetic samples with 7–28.3% porosity were placed into water and signals
were recorded after transmission through the samples (see Figure 1.7). Plona
was able to directly identify the slow P -wave, reporting propagation velocities
around 1000 m/s. Seismograms of recorded signals for varying angle of incidence
θ are shown in Figure 1.8. For normal incidence (θ = 0, Figure 1.8a), no P -to-S-
conversion occurs and only fast and slow P -waves and multiples are recorded. For
non-normal incidence, an additional converted S-wave is observed. If the angle
of incidence exceeds the critical angles of fast P - and S-waves (θ > θS c , Figure
1.8d), the seismogram is dominated by the signal of the converted slow P -wave.
In natural rocks, the slow P -wave has not been directly observed due to
their low porosity and strong microscale heterogeneities. This leads to a strong
attenuation of the slow P -wave and makes its direct detection impossible. There
is, however, indirect evidence for the existence of the effects caused by the Biot
slow P -wave. It can be shown that at frequencies below the critical Biot frequency
ωB, the slow P -wave describes a diffusion process, that influences the attenuation
and dispersion behaviour of porous rocks.
As an example, Figure 1.9 shows attenuation measurements of a partially
saturated sandstone (Murphy, 1982). Murphy applied a resonant bar technique
to obtain the frequency-dependence of partial saturation. While attenuation of
the dry sample is very low, maximum measured P -wave attenuation is as high
as 1/Q = 0.1 for 90-92% water saturation. The attenuation of shear waves is
lower and attains 0.075. The saturation-dependence of acoustic attenuation can
be explained by the effects of wave-induced local flow as described in the previous
section 1.2.
For a better understanding of fluid-related attenuation and other seismic sig-
natures the scales of the underlying process have to be analysed in more detail.
Therefore, in recent years, an increasing effort has been made to investigate the
10 Introduction
experimental setup of Plona.
Ultrasonic wave refraction at
different interfaces (a) and
an overview of compressional
wave mode multiples occur-
Figure 1.8: Seismograms
second slow compressional
(a) θ = 0,
c , and
Figure 1.9: Frequency-dependent attenuation measurements in partially
saturated sandstone conducted by Murphy (1982).
meso- and microstructure of various rock samples in the laboratory. For that
purpose, modern x-ray computer tomography (CT) is applied to estimate poros-
ity and to characterise the pore space geometry down to the micrometre scale
(e. g. Klobes et al., 1997). An even higher resolution can be obtained by neutron
radiography (de Beer et al., 2004). Commercial CT scanners commonly used in
medical radiology have resolutions in the order of millimetre and are not able to
resolve the pore space of a porous rock sample. They may be applied instead to
characterise mesoscale heterogeneities.
An example of the application of CT scans in rock physics research is given
in Figure 1.10. The figures show the development of gas patches within a water-
saturated limestone sample during a gas injection experiment. Initially, the sam-
ple is fully saturated (upper left subfigure), injection point is in the lower left
side of the rock sample. Interestingly, there is no clear gas front visible, but
gas and water form a complex patch geometry. Therefore, the scans demonstrate
that mesoscopic patchy saturation may occur during fluid replacement. The total
sample diameter is 5cm.
A combined investigation of ultrasonic velocities and CT imaging of rock
heterogeneity has been recently conducted by Monsen and Johnstad (2005) and
earlier also by Cadoret et al. (1995). They found that there is a qualitative link
12 Introduction
tially fully water-saturated
pockets (indicated by black
The image scans have a min-
imum pixel size of 0.36mm.
From Muller et al. (2008).
between the frequency-dependent dispersion characteristics of ultrasonic waves
and the patch distribution of partially saturated rocks. Lebedev et al. (2009)
showed that the speed at which the samples are saturated may influence the
mesoscopic fluid distributions and therefore affect acoustic response. Measured
seismograms at different stages of their saturation experiment are shown in Figure
1.11 together with the picked velocities.
The 3-D imaging of rocks from the pore scale to larger scales representing
whole samples is a relatively new branch of applied geophysics and sometimes re-
ferred to as digital core technology. The general availability of high-resolution
measurements of core structure motivates the development of theoretical ap-
proaches as well as numerical modelling techniques that allow to simulate the
acoustic response of real rocks on the basis of scanned images. An example
demonstrating the applicability of poroelastic finite-difference simulations for this
purpose is given in section 5.5.
1.4 Motivation and overview of this thesis
The motivation to develop a new finite-difference (FD) implementation of Biot’s
equations of dynamic poroelasticity is threefold. Actually, several FD schemes
have been presented in the past (Zhu and McMechan, 1991; Dai et al., 1995;
Jianfeng, 1999, and others, see section 3.1), but the frequency dependence and
characteristic scales were not analysed adequately by the authors, as pointed out
e. g. by Gurevich (1996). Therefore, the first objective of this thesis is to care-
fully analyse the accuracy and scalability properties of poroelastic finite-different
schemes, which is done by conducting several fundamental benchmark tests within
1.4 Motivation and overview of this thesis 13
Figure 1.11: (a) Experimentally obtained ultrasonic velocities in a par-
tially saturated rock sample. During one experiment, the sample is sat-
urated with water and the numbers indicate the stage of the saturation
experiment. (b) Signals corresponding to the five stages of saturation.
From Lebedev et al. (2009).
the frequency band from seismic to ultrasonic. Additionally, the question of nu-
merical stability under strongly heterogeneous conditions is addressed by intro-
ducing rotated FD operators that were formerly used only for FD modelling of
the elastic wave equation (Saenger et al., 2000).
Secondly, many authors analyse the influence of material properties on wave-
field attributes such as attenuation using the spectral ratio method or the fre-
quency shift method (e. g. Helle et al., 2003; Carcione et al., 2003; Picotti et al.,
2007). Although this approach is potentially very accurate, the simulation of
the underlying wavefields in computationally very expensive. As an alternative
to these classic methods, this thesis follows and further develops the ideas of
Masson and Pride (2007) and adopts the quasistatic approach to efficiently and
accurately infer dispersion and attenuation estimates for heterogeneous media.
This part of numerical applications is complemented by elastic scattering exper-
iments and quasistatic finite-element modelling.
Finally, as already mentioned above, FD modelling of poroelastic wave propa-
gation is motivated by the emergence of new laboratory experiments that allow to
characterise the details of rock micro- and mesostructure in the context of digital
core technology. In combination with physical laboratory experiments, numerical
tools may become a powerful simulation tool within the “numerical rock physics
lab”.
14 Introduction
This thesis is structured as follows. In chapter 2, the mathematical models
describing wave propagation in porous media are presented. This includes an in-
troduction to Biot theory, the governing equations, constitutive relations, plane
wave solutions for waves propagating in homogeneous media and the formulation
of boundary conditions. The chapter contains theoretical estimates for the ef-
fective properties of heterogeneous porous media and introduces different models
for the quantitative description of wave-induced fluid flow.
If theoretical solutions are not available, approximate solutions can be ob-
tained by using numerical tools. In particular, a new finite-difference scheme is
presented that allows to numerically solve the Biot equations of dynamic poroelas-
ticity in heterogeneous media (chapter 3). The stability conditions are reviewed
and the problem of numerical stiffness is introduced. It is shown how the FD
code is parallelised. Finally, a short introduction is given to the solution of con-
solidation problems using the finite-element (FE) method.
A detailed analysis of the accuracy properties of the finite-difference scheme
is presented in chapter 4. By means of fundamental examples, the applicability
of FD method is demonstrated. The obtained numerical results are compared to
exact theoretical solutions in order to estimate the approximation error. By a
scaling test the parallel performance of the numerical FD solver is checked.
Chapter 5 deals with applying numerical tools for analysing the behaviour of
heterogeneous porous media. The scattering from discrete inclusions illustrates
the conversion of different wave modes, in particular from fast to slow P -waves.
The quasistatic behaviour of synthetic heterogeneous rocks is analysed in order
to infer dispersion and attenuation characteristics from relaxation experiments.
This is the only class of problems that is based on FE modelling. Then, the focus
is on P -wave scattering experiments in random elastic as well as poroelastic media
and finally, a ultrasonic laboratory experiment is numerically simulated.
Each chapter contains a discussion of the presented material and the thesis is
finalised by concluding remarks in chapter 6.
Chapter 2
Mathematical models for wave
propagation in porous media
The propagation of elastic waves in porous media have first been described by
Biot in the 1950s as a system of two coupled wave equations. So far, preced-
ing work had focused on effective properties and consolidation of porous solids
(Terzaghi and Frohlich, 1936; Biot, 1941; Gassmann, 1951). Biot’s works on
porous media extend these results by including intertial effects to the mechanical
description and predict three distinct wave modes. In addition to the P - and
S-wave commonly known for elastic media, a second so-called slow P -wave exists
in poroelastic media. In many publications, the two compressional waves are also
referred to as type-I (fast P ) and type-II (slow P ) waves, respectively.
In order to derive the equations of motion for porous media, Biot (1956a)
assumes that continuum mechanics are applicable to the two-phase medium of
a solid matrix, saturated with a fluid. He postulates the existence of strain
and dissipation potentials and then uses Hamilton’s principle to derive the gov-
erning equations of motion. Newer works aim at establishing a more rigorous
derivation of the equations of motion, based on the clear mechanical first prin-
ciples on the microscale and using the homogenisation theory (e. g. Levy, 1979;
Burridge and Keller, 1981) or the volume-averaging method (e. g. Pride et al.,
1992).
Since the equations of motion are well-established and subject of several re-
views and text books (Attenborough, 1982; Bourbie et al., 1987; Coussy, 1991;
Carcione, 2001), the derivation will not be repeated here. Instead, the main
assumptions of the Biot theory are worked out in the following, some analytical
solutions are presented and special cases are considered. In particular, it is shown
that the theory is consistent with the elastic wave equation, with the coupled pore
pressure diffusion equation and with Gassmann’s fluid substitution relation.
An overview of the fundamental concepts of porous media is given in Figure
15
16 Mathematical models for wave propagation in porous media
Figure 2.1: Overview of theoretical descriptions of porous media.
Gassmann theory allows to calculate the effective moduli of an undrained
fluid-saturated medium. The diffusion-type interaction of pore fluid flow
with elastic deformation is described by the theory of consolidation. Addi-
tionally, inertial effects are considered in Biot theory. The associated fre-
quency regimes are commonly referred to as the static (or elastic) regime,
the quasistatic (or diffusive) regime and the dynamic frequency regime.
2.1. Furthermore, the concept of wave-induced fluid flow is introduced. The pre-
sentation includes classical theories such as the White theory of partial saturation,
but also newly developed so-called continuous random media models.
2.1 Notation
Tensor notation is used throughout this text. The components of a vector b are
written bi, cij are components of the second-rank tensor c. Since there is no
possible ambiguity, the terms vector and tensor are used for their respective com-
ponents, as well, e. g. vector bi instead of vector components bi. Conventionally,
summation over repeated indeces is carried out.
3 ∑
to time t are written as
∂φ
∂2φ
2.1 Notation 17
(gradφ)i = ∂φ
∇2φ = ∂2φ
∂xi ∂xi
where ijk is the Levi-Civita-symbol. It is defined as
ijk =



1, if (i, j, k) ∈ (1, 2, 3), (2, 3, 1), (3, 1, 2),
−1, if (i, j, k) ∈ (1, 3, 2), (3, 2, 1), (2, 1, 3),
0, else.
(2.8)
The Kronecker symbol δij is also used as the equivalent of the unit tensor 1
δij =
0, if i 6= j . (2.9)
A similar symbol is used for the Dirac distribution δ(t). It is related to the
Heaviside step function, both are defined such that
δ(t) = 0 ∀t 6= 0 with
∫ ∞
−∞
1 for t ≥ 0 . (2.11)
If a Fourier transform is required, it is written using the symbol F and trans-
formed quantities from the time domain to the frequency domain are indicated
by a tilde
F {φ(t)} = φ(ω) . (2.12)
The kinematic field variables used in the present context are the displacements
of the solid frame ui and the displacements of the fluid phase uf i . Relative dis-
placements wi are defined as
wi ≡ φ(uf i − ui) , (2.13)
where porosity φ is the volume fraction of the pore space. Strain of the solid
matrix εij is related to the displacements via the kinematic relation
εij ≡ 1/2 ( ∂jui + ∂iuj) , (2.14)
18 Mathematical models for wave propagation in porous media
its trace or the divergence of the solid displacement is denoted as
ε ≡ εii = ui,i , (2.15)
and that of the relative displacement is called the increment of fluid content
ζ ≡ −wi,i . (2.16)
The formulation of the governing equations using the relative displacement in-
stead of the fluid displacement was introduced by Biot (1962). The present work
follows closely the modern presentation of the textbook by Carcione (2001).
2.2 Momentum equations
Biot’s linear theory of poroelastic wave propagation is valid under the following
assumptions: (i) only connected pores are considered in the equations and dis-
connected pores are treated as part of the solid matrix, (ii) the porous medium
is statistically isotropic, i. e. porosity and permeability are the same in all direc-
tions, (iii) the wavelength is large compared to the microscopic porescale and (iv)
deformations are small in order to ensure linear elastic material behavior.
Then, neglecting source terms, Biot’s equations for an isotropic fluid saturated
porous medium are given by
ρbui + ρf wi = ∂j τij (2.17)
ρf ui + Y ∗ wi = −∂i p. (2.18)
On the right hand side of these vector equations, the divergence of the total stress
field τij and the gradient of pore pressure p appear. They are discussed later in
section 2.3. Now on the left hand side, four intertial terms are given, with the
bulk density ρb determined from the density of the solid grains ρs and that of the
pore fluid ρf by
ρb = φρf + (1 − φ)ρs. (2.19)
The viscodynamic operator Y is a function of the differential operator ∂t, and
in the frequency domain it becomes a complex, frequency-dependent quantity
(Biot, 1956b). Biot evaluates the oscillatory flow in a circular duct as a model for
a porous solid and expresses the viscodynamic operator with the help of Bessel
functions. Johnson et al. (1987) use the concept of dynamic permeability k(ω)
to introduce the frequency dependence of the operator, i. e.
Y = η
k(ω) = η
2.3 Constitutive relations 19
Here, η is the dynamic viscosity of the pore fluid, κ is the dc permeability of the
porous matrix, ωB is the critical transition frequency and n is a dimensionless
parameter that is related to size of the pore channels. The frequency ωB plays an
important role in the characterisation of the mechanical regime for homogeneous
porous solids, since for frequencies lower than ωB, the flow behaves laminar and is
of Poiseuille type. However, for frequencies exceeding ωB, deviations occur from
the laminar flow and therefore additional parameters are needed to characterise
the dynamic behaviour of the flow field and of the corresponding mechanical
response of the porous composite. The critical frequency is calculated according
to
ωB ≡ η φ
κ νρf , (2.21)
where ν refers to the tortuosity of the pore space, a dimensionless number larger
or equal to one. Now, inserting equation 2.21 into equation 2.20 and taking the
limit of n→ ∞ results in
Y = ρfν
κ = ρm ω + b. (2.22)
The quantities ρm and b are referred to as effective fluid density and the hydraulic
friction coefficient, respectively. They are given by
ρm = ρfν
φ , (2.23)
b = η
κ . (2.24)
The simple form of the operator Y given in equation 2.22 is referred to as the
classical low-frequency approximation as used in Biot (1956a). The expression
consists of an inertial part ρmiω and a viscous term b, the latter being responsible
for internal friction between the pore fluid and the solid frame. Casting equation
2.22 into the momentum equation 2.18 yields the low-frequency formulation of
the momentum equations for porous media
ρbui + ρf wi = ∂j τij (2.25)
ρf ui + ρmwi = −∂i p− bwi . (2.26)
In the chapter on numerical methods, this formulation of the momentum equa-
tions is usually referred to.
2.3 Constitutive relations
Poroelastic constitutive laws relate the total stress field τij and the pore pressure
p to the deformation state of a porous medium. The two independent deformation
20 Mathematical models for wave propagation in porous media
fields are εij and ζ as defined in equations 2.14 and 2.16, respectively. With the
help of these strain variables, the poroelastic constitutive relations are written in
the general, linear case (Carcione, 2001)
τij = cuijklεkl − αijMζ , (2.27)
p = −αijMεij +Mζ . (2.28)
The three material parameters in these equations are the undrained elasticity
tensor cuijkl, the tensor of effective stress coefficients αij and the so-called pore
space modulus M . If the medium is isotropic, cuijkl can be expressed via the two
Lame parameters λu and µ. The tensor αij then reduces to a scalar, such that
cuijkl = λuδijδkl + µ (δikδjl + δilδjk), (2.29)
αij = α δij. (2.30)
Introducing equations 2.29 and 2.30 into the relations 2.27 and 2.28, the isotropic
constitutive relations are obtained as
τij = 2µεij + λu ε δij − αM ζ δij, (2.31)
p = −αMε+Mζ. (2.32)
In order to illustrate the meaning of these relations, one might consider a few
special deformation states and introduce 6 fundamental poroelastic moduli. Be-
ginning with pure shear and pure dilatational deformation under undrained con-
ditions ζ ≡ 0, one obtains expressions for the undrained shear and bulk moduli.
Using the deformation angle γij = 2 εij for i 6= j, they are
Gu ≡ τij γij
3 µ. (2.34)
The same two deformations are now applied using drained conditions with p ≡ 0.
In the pure shear case, ε = 0 and p = 0 imply ζ = 0 and therefore
Gd ≡ τij γij
= 2µ εij
2 εij
= µ. (2.35)
In the case of pure dilatation, equation 2.32 provides ζ = αε and if this is sub-
stituted into equation 2.31 one computes the drained bulk modulus Kd as
Kd ≡ τii 3 ε
Figure 2.2: Sketches of three deformation experiments for the determi-
nation of the drained and undrained bulk moduli Kd and Ku (a,b) as well
as the shear modulus G (c) that is independent of the fluid properties.
By comparing the results for the undrained and the drained one obtains easily
the famous Gassmann result (Gassmann, 1951)
Gu = Gd = G, (2.37)
Ku = Kd + α2M, (2.38)
that is that the shear modulus is not affected by the presence of fluid in the
pore space and that the undrained bulk modulus is easily obtained from the
drained modulus by adding α2M . The three corresponding experiments for the
determination of Kd, Ku and G are shown in Figure 2.2. By means of a simple
gedankenexperiment (Biot and Willis, 1957; Brown and Korringa, 1975), α and
M can furthermore be related to the bulk moduli of the solid grains Kg and of
that of the pore fluid Kf :
α = 1 −Kd/Kg, (2.39)
Eventually, drained and undrained uniaxial strain conditions provide two vertical
incompressibilities, Pd and Pu, that are also denoted as L and H, respectively.
Since they are closely related to the velocity of P -waves, they are also called
drained and undrained P -wave moduli. Without derivation, they are given as
Pd = L ≡ τzz
= Kd + 4/3G, (2.41)
Pu = H ≡ τzz
= Ku + 4/3G. (2.42)
2.4 Plane wave solutions
A system of coupled linear wave equations for the displacements ui and wi is
obtained by inserting the constitutive relations 2.31 and 2.32 into the momentum
equations 2.17 and 2.18, so that
ρbui + ρf wi = (λu + µ)uj,ji + µui,jj + αM wj,ji , (2.43)
ρf ui + Y ∗ wi = αM uj,ji +M wj,ji . (2.44)
Using the vector theorem
ui,jj = uj,ji − ijkklmum,jl (2.45)
and substituting the poroelastic moduli H = λu + 2µ as well as G = µ and
C = αM , the wave equations become
ρbui + ρf wi = H uj,ji + C wj,ji −Gijkklmum,jl (2.46)
ρf ui + Y ∗ wi = C uj,ji +M wj,ji. (2.47)
On the right hand side of equations 2.46 and 2.47, the spatial derivatives grad
div and rot rot of the displacement fields appear. Now, the Helmholtz theorem
states that any vector field can be decomposed into the sum of an irrotational
and a solenoidal vector field. This means that for the irrotational part of the
displacement field, the contribution from the third term on the right hand side
of equation 2.46 disappears. At the same time, for the solenoidal part all the
terms that contain the divergence operator vanish. As in the case of elastic
wave propagation, compressional and shear waves are therefore decoupled. The
dispersion relation of all wave modes are obtained by using plane waves as an
ansatz for the solution of equations 2.46 and 2.47.
A plane wave propagating in direction x with wavenumber k and circular
frequency ω has the form
u = u0 exp[ (kx− ωt)], (2.48)
where u(x, t) = (u, w) and u0 = (u0, w0) is constant. Inserting this ansatz into
the wave equations and assuming irrotational motion, one finds the following
equation in matrix form
where the matrices P and H are given by
P =
and
H =
. (2.51)
This is an eigenvalue problem with the unknown eigenvalues (k/ω)2. They are
calculated as the solution to the characteristic equation
det (
D = H−1P (2.53)
one obtains explicitly the dispersion relation for plane P -waves as
k2
ω2 =
1
2
(2.54)
with
trD = 1/ detH [
. (2.58)
The same reasoning leads to a characteristic equation in the case of purely
solenoidal particle motion. In that case one has
det (
G = − (
. (2.60)
Due to the irregular but simple form of G, the dispersion relation for S-waves is
k2
ω2 = [
tr (
P−1G )]−1
= − detP ω
Y G . (2.61)
So far, the two roots of the characteristic equations for compressional waves and
the third root of that for shear waves correspond to the three wave modes in
porous media. The compressional waves are referred to as fast and slow P -waves
or sometimes waves of the first and second kind, respectively. The fast P -wave
behaves similarly to the compressional wave mode in elastic media, which is why
it often simply referred to as the P -wave. The slow P -wave is a particularity
of poroelastic media and it usually strongly attenuated in real porous rocks.
24 Mathematical models for wave propagation in porous media
10 −4
10 −2
10 0
10 2
A tt e n u a ti o n 1
/Q
(b)
P1
S
P2
Figure 2.3: Dispersion (a) and attenuation (b) of the three wave modes
in a porous medium (water saturated sandstone, see Tables 2.1 and 2.2).
Fast P - and S-wave show very small dispersion, while slow P -velocity tends
to zero at low frequencies. Note that the inverse quality factors of P1 and
S in (b) have been multiplied by 50.
Actually, at frequencies below the critical Biot frequency ωB, this wave mode
becomes diffusive, while at frequencies higher than ωB, it is a propagating wave
mode. Due to the diffusive behaviour at low frequencies, the slow P -wave is often
considered as a diffusion wave as discussed in more detail in section 2.6.
Propagation velocities v and attenuation in the form of the quality factor Q
are calculated from the wavenumbers as
v = ω
Re k . (2.63)
Dispersion curves as well as attenuation behaviour for compressional and shear
waves are given in Figure 2.3. The medium considered is water-saturated con-
solidated sandstone model with material parameters given in Tables 2.1 and 2.2.
The dispersion curves in Figure 2.3a show a very small frequency dependence of
the fast P - and S- waves. Both are higher than the velocity of the slow P -wave,
of which the velocity tends to zero as frequency decreases. On the other hand, the
attenuation of the slow P -wave at frequencies below ωc tends to one, see Figure
2.3b. Attenuation values for the S- and P -waves is much smaller; for convenience,
the attenuation amplitudes have been enlarged by a factor of 50.
2.5 Boundary conditions
In order to fully specify a particular poroelastic problem, boundary conditions
and initial conditions are required to constrain the solution of the governing
2.5 Boundary conditions 25
equations. For wave propagation problems, the initial conditions are usually a
stress-free medium at rest, i. e. all field variables are 0. Generally, the initial con-
ditions must fulfil all the governing equations and they must be consistent with
the boundary conditions. Deresiewicz and Skalak (1963) considered the possi-
ble cases for external and internal boundaries. A traction free boundary at the
edge of the considered problem domain is described by setting the normal stress
component and the pore pressure to 0 as
τij nj = 0, p = 0 , (2.64)
with nj being the unit vector in normal direction. Similarly, setting the displace-
ments to 0 gives fixed boundaries with
ui = 0, wi ni = 0 . (2.65)
Obviously, boundary conditions of mixed type, e. g. a mechanically constrained
but drained condition or a unconfined/undrained condition are also possible. If
two porous media are in welded contact with each other, continuity is required
for traction, pore pressure, total displacement and the normal component of the
relative displacement. Sometimes, a mixed boundary condition is applied for the
pore pressure to account for partial closure of the pore space at the interface. This
is done by introducing a surface resistance parameter k, where k = 0 corresponds
to the standard case of an open internal boundary. Denoting the two sides of the
interface by a superscript, the porous-porous boundary conditions then read
u1 i = u2
i ni, τ 1 ij nj = τ 2
ij nj, p1 − p2 = k wini . (2.66)
Eventually, the two special cases of a porous medium in contact with a fluid or
with an elastic solid are given. For a porous-solid interface one has
u1 i = u2
ij nj = p1 = τ 2 ij nj , (2.67)
and for a porous-fluid interface
(u1 i + w1
ii/3 = p1 = p2 . (2.68)
On the basis of these boundary conditions, reflection and transmission coeffi-
cients of waves in porous media are derived in works by Deresiewicz and Skalak
(1963) for reflections at an external boundary and by Dutta and Ode (1983) for
a gas-water interface. Simplified expressions for normal incidence are given in
Gurevich et al. (2004).
Table 2.1: Typical material properties of a consolidated and unconsoli-
dated sandstone.
Shear modulus G GPa 11.0 1.0
Porosity φ 0.2 0.3
Tortuosity ν 2.0 2.0
Permeability κ 10−12m2 1 1000
Grain density ρg kg/m3 2650 2650
2.6 Quasistatic behaviour of porous media
Many mechanical processes in porous media are slow in the sense that the intertial
terms in the momentum equations become negligible. The temporal change in the
displacement and stress fields is then only driven by the internal friction between
the grain framework and the pore fluid. In this case, the viscodynamic operator
reduces to the friction coefficient b = η/κ and the momentum equations 2.43 and
2.44 become
b wi = −p,i . (2.70)
These two equations are an equilibrium condition for the total stress field τij and the equation of Darcy flow stating the proportionality between the negative
pressure gradient ∂ip and the filtration velocity wi. A different way of presenting
the equilibrium and Darcy’s law is obtained by taking the divergence of the Darcy
equation and substituting the constitutive relations. Following Wang (2000), one
finds an equilibrium condition expressed in terms of the displacement field ui
Table 2.2: Material properties of fluids typically found in reservoirs (wa-
ter, gas, oil) or in the laboratory (air) (Batzle and Wang, 1992).
unit water oil gas air
Bulk modulus Kf GPa 2.25 1.3 0.1 0.00014
Density ρf kg/m3 1000 850 100 1
Viscosity η mPas 1.0 4.0 0.22 0.02
2.6 Quasistatic behaviour of porous media 27
coupled with an inhomogeneous diffusion equation for the pore pressure p as
0 = (λd + µ)uj,ji + µui,jj − α p,i (2.71)
α ui,i + p/M = p,ii/ b . (2.72)
Equations 2.71 and 2.72 describe the coupled quasistatic deformation and pore
pressure diffusion in porous media, such as e. g. consolidation processes of fluid
saturated soils or reservoir compaction during depletion. They imply that in the
general case, the pore pressure field cannot be calculated by a single diffusion
equation alone, but the influence of temporal changes in pore pressure on the
overall stress and deformation field (and vice versa) needs to be accounted for.
In four specific circumstances, the pore pressure equation, however, uncouples
from the equilibrium equation and can therefore be solved independently (Wang,
2000). These circumstances are“(1) steady state, (2) a state of uniaxial strain and
constant vertical stress, (3) a highly compressible fluid, and (4) an irrotational
displacement field in an infinite domain without body forces” (Wang, 2000).
For the four cases, simple analytical solutions are available. In particular,
under assumption (4), the diffusion equation 2.72 is decoupled from the strain
term and becomes a homogeneous diffusion equation
p/N = p,ii/ b (2.73)
with the poroelastic modulus
N = MPd
H . (2.74)
The quotient of b and N is called the hydraulic diffusivity
D = N/ b = κMPd
η H (2.75)
and it is the main material parameter describing diffusion problems. In the
unidimensional case, the solution of equation 2.73 due to an instantaneous point
source with p(x, t = 0) = δ(x) is (Rudnicki, 1986)
p(x, t) = 1√
4Dt (2.76)
Now, if the point source is a Heaviside step function in time p(x = 0, t) = H(t),
the unidimensional pore pressure response is expressed as
p(x, t) = erfc x√ 4Dt
. (2.77)
Some pore pressure profiles resulting of a point source in a 1-D medium are given
in Figure 2.4.
0 1 2 3 0
0.05
0.1
0.15
0.2
0.2
0.4
0.6
0.8
1
(b)
Figure 2.4: Quasistatic pore pressure response due to a point source
in a unidimensional homogeneous porous medium (water saturated con-
solidated sandstone, see Tables 2.1 and 2.2). Profiles are given for (a) an
instantaneous source and (b) a constant pressure source.
Eventually, the diffusion equation is applied to derive an low-frequency ap-
proximation of the slow P -wavenumber that was already given in equation 2.54.
Introducing the plane wave ansatz into equation 2.73 gives the dispersion relation
for a “pure diffusion wave” k2
ω2 =
b
ωN . (2.78)
Simple approximations are obtained for the fast P -wave and for the shear wave,
as well, by considering that at low frequencies with ω ωB the dispersion and
attenuation are very small. Actually, at low frequencies, friction between the
fluid and the grain matrix is dominant and the porous medium is practically
undrained. Setting the relative motion wi to zero, the coupled wave equation
2.46 reduces to a purely elastic wave equation
ρbui = H uj,ji −Gijkklmum,jl (2.79)
from which dispersion relations for the fast P - and S-waves are derived as
k2
ω2 = ρb
G . (2.81)
Figure 2.5 shows a comparison of predicted wave propagation velocities by the
full formulas 2.54–2.61 and the approximations 2.78, 2.80 and 2.81. As is demon-
strated there, the simplified formulas provide a good approximation of the full
wavenumbers at frequencies sufficiently below the critical Biot frequency ωB.
2.7 Heterogeneous porous media 29
10 −4
10 −2
10 0
10 2
A tt e n u a ti o n 1
/Q
(b)
n → ∞
qsa
Figure 2.5: Dispersion velocities and attenuation of the slow P -wave
in poroelastic media. The quasistatic approximation (qsa) is compared to
the full formulation with dynamic operators according to equation 2.20.
2.7 Heterogeneous porous media
So far, only waves in homogeneous porous media have been considered, where
homogeneous refers to all scales larger than the porescale. For the homogeneous
case in an unbounded domain, plane wave solutions as well as Green’s functions
for instantaneous point sources are available (See Karpfinger, 2006, for a review).
In contrast, for heterogeneous media, i. e. if the medium properties vary as a
function of position, general solutions are available only for a few special cases.
Often, the properties of the heterogeneous medium are expressed based on the
statistical properties of the single constituents. An average of a quantity φ over
the representative elementary volume is then denoted as
φ = 1
φ(x) dV . (2.82)
If a mixture of two or more fluid phases fill the pore space of a homogeneous rock,
one speaks of partial saturation. In contrast, a double porosity model consists of
a heterogeneous rock fully saturated with only one fluid phase. Partial saturation
occurs typically in a reservoir where one fluid, e. g. oil is replaced by another fluid
such as water during production. Usually, the first fluid phase is not completely
replaced by the second fluid, but both may form patches on multiple scales. An
example of double porosity media is a fractured reservoir in which cracks and
fractures can be considered as soft and permeable inclusions within a stiff host
rock.
A particularity of partial saturation is that exact theoretical limits are avail-
able for the estimation of effective elastic moduli. This estimate is based on the
assumption that a propagating compressional or shear wavelength is much larger
than the scale of the fluid patch. The first result is that the effective shear modu-
30 Mathematical models for wave propagation in porous media
lus is not affected by the presence of different fluids in the rock, which is a direct
consequence of the solenoidal character of the shear motion and the constitutive
relations 2.31 and 2.32. Now, if a compressional wave propagates through a par-
tially saturated medium, a fluid diffusion process is induced and the spatial scale
of this process is estimated as
λD =
ω , (2.83)
where the approximate diffusivity D = Kf/b is the governing hydraulic parame-
ter. λD is sometimes called diffusion length (Norris, 1993). If it is small compared
to the scales of the fluid patches, no internal fluid flow occurs, the system is unre-
laxed and behaves like a heterogeneous elastic medium with varying bulk modulus
but constant shear modulus. In this particular case, the theorem of Hill (1963)
states that the heterogeneous medium is effectively isotropic with a bulk modulus
Keff is determined as the weighted harmonic average of the individual bulk mod-
uli, independent of the distribution of the fluid phases inside the volume. One
writes
, (2.84)
where the individual moduli K are calculated using Gassmann’s equation. This
limit is referred to as Gassmann-Hill.
Now, if the diffusion scale is much larger than the spatial scale of the sys-
tem, the pore pressure is equilibrated throughout the medium and the effective
modulus is obtained by substituting a harmonically averaged effective fluid bulk
modulus into Gassmann’s equation
Kf eff =
yielding the so-called effective fluid model or Gassmann-Wood average (Wood,
1955). In Figure 2.6, the theoretical limits are given for two fluid mixtures (water–
gas and water–air) that saturate a consolidated sandstone, for material properties
see Tables 2.1 and 2.2. Both curve pairs show qualitatively the same behaviour:
an approximately linear increase in P -wave velocity from fully gas- to fully water-
saturated is characteristic for the Gassmann-Hill bound, while the Gassmann-
Wood bound shows a very strong decay of the effective bulk modulus when only
a little gas is introduced into a fully water-saturated medium. The decay is the
more pronounced, the stronger the contrast is between the fluid phases.
The criterion 2.83 has been given to distinguish between the unrelaxed state
(Hill) and the relaxed state (Wood). The limits can as well be considered as the
2.7 Heterogeneous porous media 31
0 0.2 0.4 0.6 0.8 1 4
6
8
10
12
s K
e ff (
G P
water and gas, or water and
air, respectively. The curves
ulus Keff as function of water
saturation.
high- and low-frequency limits of partial saturation models, where a crossover
occurs around the critical frequency
ωc = D
L2 . (2.87)
Unfortunately, the Hill theorem does not apply to an elastic medium with varying
shear modulus. This implies that theoretical bounds for double porosity media
are in general not available. An exception is the case when the porous ma-
trix is a conglomerate of only two porous phases that are distributed such that
the conglomerate is again isotropic. Then, effective bulk moduli for the grains
and for the mat